As an Adjunct Mathematics Instructor at Olympic College, Dylan Lasher uses R to simulate probability problems like the Birthday Paradox that are a bit more complex than standard calculations. Students are directly engaged through hands-on coding and experimentation in R, rather than passively receiving information. They create simulations, run trials, visualize outcomes, and dynamically explore probabilistic concepts, thus becoming deeply embedded in their own learning process. Here’s a clear outline of how Lasher implements an activity exploring the Birthday Paradox with his introductory statistics students.
- Introduction and Intuition Check
First, students individually guess how large a group must be before two people likely share a birthday. They discuss briefly in pairs, making initial informal predictions. - Demonstration (Slow Trial)
The class runs one or two trials together slowly on the screen using a simple R snippet (which Lasher provides) to generate random birthdays and stop once a match is found. Students immediately see how the trial process works step-by-step. - Theoretical Calculation
Next, the class formally calculates the theoretical probability (e.g., that about 50% probability of having a birthday match occurs around 23 people). Students now refine their initial predictions using this theoretical knowledge. - Fast, Large-Scale Simulation
Students then run tens of thousands of trials using the provided starter code in R (usually a function they write as a class, logic step by logic step). This function generates thousands of trials rapidly, recording how many people were needed before a match was found each time. Students then visualize the distribution using simple histograms or boxplots. - Making Sense of Data and Results
Students write brief conclusion statements comparing their theoretical calculations to the simulation outcomes. A common prompt is: “How does your theoretical prediction compare to your simulated average? Why might differences occur?” This helps students see clearly how theory aligns with practice and emphasizes random variability. - Deeper Discussion and Reflection
Finally, we use guided discussion prompts consistently each semester, such as:- “Why is your intuition often different from reality here?”
- “Where else in real life might this surprising probability show up?”
- “How could misunderstanding these probabilities impact important decisions?”
Digital Resources
R and RStudio: free software for statistical computing and data analysis
Students often predict the wrong answer to non-intuitive probability problems like the birthday paradox, making the visual result from digital simulation striking and memorable. Math doesn’t tend to have a “lab” section, so simulations help students make a theoretical prediction and then see repeated simulations aggregate to their prediction. This helps build probabilistic intuition. This activity is especially effective for students with a disconnection of math from “real world” application. Too many students have experienced problem solving where they get a number, write it down, and move to the next question without considering the context of the number they wrote. Engaging with code rather than abstract formulas allows students to explore ideas safely and concretely.
- Code Sharing: Lasher usually provides the base R code to ensure equity in access, allowing students to tweak parameters rather than start from scratch. These are all posted online after class.
- Real-World Connections: Lasher explicitly prompts students to reflect on real-life scenarios to enhance the relevance and instructional transparency.
- Student Reflections: At the end of the activity, students write a short reflection paragraph synthesizing their predictions, theoretical calculations, simulation results, and discussions, reinforcing the practice of metacognitive thinking.
Throughout the activity, formative assessment occurs organically. Students first predict outcomes based on intuition, then compute theoretical probabilities, and finally compare these predictions with their simulation results. Immediate feedback from R’s simulation allows students (and Lasher) to quickly identify and correct misconceptions during class itself. The post-activity discussion explicitly connects classroom probability concepts to real-world scenarios that students care about (such as cybersecurity, genetic matching, or medical testing), illustrating the importance of probability beyond the classroom. This transparency helps students understand exactly why the topic matters and how the concepts learned can apply to their lives and future careers. Through informal verbal assessments of students, Lasher sees increased conceptual confidence and satisfaction with applications post-activity.
Digital Enablement
The real benefit of using R for probability simulations like the Birthday Paradox is precisely the ability to rapidly execute thousands of trials, ensuring genuine randomness and meaningful aggregation of results. If students were asked to manually generate birthdays (1) they would inevitably discover that humans are pretty awful at choosing truly random values (as demonstrated from a demonstration of flipped coins versus made-up coin flips, where you can easily tell the difference in patterns), and (2) the exercise would quickly become impractical and tedious. By using R, students are likely to trust the randomness more, because R transparently generates data using established/vetted statistical procedures. Running many trials quickly in R reinforces students’ theoretical predictions by showing empirical convergence to the theoretical probabilities, making the concept far more believable, accurate, and impactful to students. In short, R gives immediate and credible empirical evidence that strengthens students’ intuitive and theoretical understanding of probabilities.
Probability can often feel abstract or exclusionary for students with weaker mathematical backgrounds. By using free software for simulations that clearly visualize outcomes, every student, regardless of prior experience, is provided with equal footing to see probability “in action.” Additionally, by framing the activity around collaborative problem-solving and sharing outcomes openly, students feel their voices and intuitions are validated, fostering a stronger sense of belonging in the statistics classroom.